Classical Analysis: An Approach Through Problems

I strongly recommend this book, a textbook,for a first-year analysis course, for students who have completed a one-year introductory sequence in Calculus. The book has a special target population: undergraduates who intend or may be getting involved in undergraduate research. With the growth of undergraduate research including journals devoted to undergraduate research, this book is a welcome tool facilitating this growth. In the author’s own words (preface), “Historically, deep and beautiful mathematical ideas have often emerged from simple cases of challenging problems. Examining simple case allows students to experience working with abstract ideas at a nontrivial level- something they do little of in standard mathematics courses.”

 

The author’s vehicle for exposing aspiring research undergraduates to the taste, flavor, challenges, frustrations, and rewards of research, is the inclusion of over 200 problems from the American Mathematical Monthly, the College Mathematics Journal, Mathematics Magazine, and the Putnam competitions. The author did not simply select and gather challenging problems; rather, the problems selected have fairly short statements, are challenging, and have pleasing conclusions, requiring at least one ingenious idea. The problems are organized by topic, simultaneously illustrating analysis principles and going beyond them showing nuances of the theorems and tools of analysis which can only be gained through problem-solving experience.

 

The author has also introduced a novel design in this textbook which I would encourage other authors to adopt. By way of background and contrast, most textbooks have numbered sections with an unnumbered section titled ”Exercises” at the end of each chapter. The classic Olmstead Calculus introduced the simple idea of numbering the exercise sections also, thereby indicating that exercises are not just for assessment, but are an intrinsic part of the course. The current textbook goes to the next level: Each chapter has two problem sections, one titled ”Worked out examples” and one titled ”Exercises.” Thus, the student is not just given tough problems; rather the student is exposed to numerous worked-out solutions in the ”Worked out examples” section and then asked to try their own hand at problem-solving in the exercise section.

 

My favorite assessment method for a Calculus or Analysis book is to go to the numerical series section and see whether it is routine or goes beyond the routine. This book does cover the bread and butter techniques of series: the comparison, ratio, geometric, p-series, root, absolute convergence, and integral tests. But the book goes beyond the routine. It also presents the Raabe, logarithmic, Kummer, Bertrand, and Cauchy Condensation tests.

 

This emphasis on extra material persists in other chapters such as the inclusion of a section on convex functions in the chapter on differentiation, a section on the Weierstrass approximation theorem in the chapter on function sequences, and a section on the Gamma function in the chapter on improper and parametric integration.

 

Despite the above, the book is not excessively long. Besides the 14 sections on ”Worked out examples” and ”Exercises” all the material is comfortably covered in the remaining 30 sections.

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Russell Jay Hendel, holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, graph theory, applications of technology to education, problem writing, theory of pedagogy, actuarial science, and the interaction between mathematics, art, poetry, and literary exegesis.